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Theorem mrcval 16093
 Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcval ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
Distinct variable groups:   𝐹,𝑠   𝐶,𝑠   𝑋,𝑠   𝑈,𝑠

Proof of Theorem mrcval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
21mrcfval 16091 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
32adantr 480 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
4 sseq1 3589 . . . . 5 (𝑥 = 𝑈 → (𝑥𝑠𝑈𝑠))
54rabbidv 3164 . . . 4 (𝑥 = 𝑈 → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
65inteqd 4415 . . 3 (𝑥 = 𝑈 {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
76adantl 481 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) ∧ 𝑥 = 𝑈) → {𝑠𝐶𝑥𝑠} = {𝑠𝐶𝑈𝑠})
8 mre1cl 16077 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
9 elpw2g 4754 . . . 4 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
108, 9syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
1110biimpar 501 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
128adantr 480 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋𝐶)
13 simpr 476 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈𝑋)
14 sseq2 3590 . . . . . 6 (𝑠 = 𝑋 → (𝑈𝑠𝑈𝑋))
1514elrab 3331 . . . . 5 (𝑋 ∈ {𝑠𝐶𝑈𝑠} ↔ (𝑋𝐶𝑈𝑋))
1612, 13, 15sylanbrc 695 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑋 ∈ {𝑠𝐶𝑈𝑠})
17 ne0i 3880 . . . 4 (𝑋 ∈ {𝑠𝐶𝑈𝑠} → {𝑠𝐶𝑈𝑠} ≠ ∅)
1816, 17syl 17 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ≠ ∅)
19 intex 4747 . . 3 ({𝑠𝐶𝑈𝑠} ≠ ∅ ↔ {𝑠𝐶𝑈𝑠} ∈ V)
2018, 19sylib 207 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → {𝑠𝐶𝑈𝑠} ∈ V)
213, 7, 11, 20fvmptd 6197 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∩ cint 4410   ↦ cmpt 4643  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070 This theorem is referenced by:  mrcid  16096  mrcss  16099  mrcssid  16100  cycsubg2  17454  aspval2  19168
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